Prey-predator model with nonlocal and global consumption in the prey dynamics

Figure 1.  Single pulse solution for prey and predator population for $ b = 10, d_1 = 0.1, d_2 = 0.1 $

Figure 2.  Moving pulse for $ b = 10, d_1 = 0.3, d_2 = 0.1 $ (a) after time $ t = 500 $; (b) after time $ t = 600 $; (c) $ x $-$ t $ profile for $ L = 20 $

Figure 3.  For two bifurcation diagrams, $ a = 1 $, $ \sigma_1 = 0.1 $, $ k_1 = k_2 = 0.35 $, $ \sigma_2 = 0.2 $ are fixed. (a) Bifurcation diagram in $ (L,d_1) $-plane for the values of parameters $ b = 5 $, $ \alpha = \beta = 0.35 $, $ d_2 = 0.1 $. (b) Bifurcation diagram in $ (d_1,b) $-plane for the values of other parameters $ \alpha = \beta = 0.363 $, $ d_2 = 1 $

Figure 4.  Periodic travelling wave in the case of nonlocal consumption followed by a spatio-temporal structure. Left: a snapshot of solution with prey (blue) and predator (red) distributions. Right: level lines of the prey distribution $ u(x,t) $. The values of other parameters are $ a = 1,b = 1, \alpha = \beta = 0.363, \sigma_1 = 0.1, k_1 = k_2 = 0.35, \sigma_2 = 0.2, d_1 = d_2 = 1 $

Figure 5.  Multiple moving pulses in the case of nonlocal consumption. Left: a snapshot of solution with prey (blue) and predator (red) distributions. Right: level lines of the prey distribution $ u(x,t) $. The values of other parameters are $ a = 1,b = 1, \alpha = \beta = 0.375, \sigma_1 = 0.1, k_1 = k_2 = 0.35, \sigma_2 = 0.2, d_1 = d_2 = 1 $

Figure 6.  Different regimes observed in the case of nonlocal consumption presented on the $ (d_1,N) $ parameter plane for $ \alpha = 0.35 $ (left) and $ (N,\alpha) $ parameter plane for $ d_1 = 1 $ (right). The values of other parameters are $ a = 1,b = 1,\sigma_1 = 0.1, k_1 = k_2 = 0.35, \sigma_2 = 0.2, d_2 = 1 $